Context

This is an exploration of the relevance to higher orderings of
the insights of the Tao Te Ching of thinking with regard to what is termed
hyperspace by mathematicians (understood here to include physicists).

This is necessarily a presumptuous, and possibly foolish, undertaking
-- since, for mathematicians, hyperspace requires consideration of at least
10 dimensions and the arduous mathematical training to understand the relevant
equations. This exploration must therefore be more of an intuitive, speculative
exercise in pointing to suggestive possibilities. There is some consolation
in that one renowned physicist, Edward Witten, argues that physics is not about
complex calculations: "The essence of it is that physics is about concepts,
wanting to understand the concepts, the principles by which the world works"

It is worth noting that several hundred international physics
conferences have been convened to explore the consequences of the higher dimensionality
associated with hyperspace.

Field of consciousness and the Tao Te Ching

The individual, in framing and dealing with reality, can be understood as being
at the centre of a field of consciousness offering a range of possibilities.
This field can be understood as organized in many ways. The Tao Te Ching
has long provided a much respected pattern of insights -- possibly to be understood
as a distillation of awareness about awareness. For the purposes of this exercise,
the focus here is on how the 81 insights of the Tao Te Ching might be
understood as ordering the range of potential modes of awareness -- both explicit
and implicit.

In the accompanying exploration of the 9-fold
Higher Order Patterning of Tao Te Ching Insights, much attention was
given to their possible disposition in a 9x9 matrix -- as an array of insights
(disposed in a peacock's tail in some cultural symbolism). Crudely this could
then be seen as constituting a kind of setting for a children's game. As with
hopscotch, for example, an individual might move from one cell to another --
with each cell being associated with a different perspective, insight or mode
of awareness. And with each cell offering different kinds of connectivity to
other cells. The challenge in that exploration was to find more powerfully integrative
ways of ordering such an array -- hence the exploration of magic squares, and
the possible relevance of mathematical objects of higher dimensionality, such
as hypercubes. The emphasis however was on how any such order was to be comprehended.

For mathematicians the exploration of hyperspace (according to the admirable
description of Michio Kaku: Hyperspace, 1994) is based on the "field"
theory originated by Faraday -- inspired by an agricultural metaphor. For him,
a field occupies a region of three-dimensional space such that at any point
in the space a collection of numbers can be assigned that describes the magnetic
or electric force at that point. In its development by Georg Riemann (1854),
a collection of numbers at every point could be introduced to indicate how much
the space was bent or curved. On a two-dimensional surface, a collection of
three numbers at every point completely described the bending of the surface
-- whereas in four spatial dimensions a collection of 10 numbers was required
at each point to describe its properties.

Riemann's metric tensor in 4 dimensions
with the information necessary to describe a curved space. In this case,
16 numbers are required to describe each point. 6 of them are redundant
(eg g12 = g21) leaving 10 independent numbers. These
can then be arranged in a square array

g11

g12

g13

g14

g21

g22

g23

g24

g31

g32

g33

g34

g41

g42

g43

g44

With this device Riemann could then describe N-dimensional space with a metric
tensor that would then resemble a chess board that was NxN in size. In the quest
to provide a unified description, the metric tensor could be expanded to N-dimensional
space then portions of it -- in the form of rectangular pieces -- could be identified
as corresponding to different forces embodied in the unified description. Whereas
Maxwell's classical field equations for electricity and magnetism are 8 in number,
these collapse into a single relativistic equation when time is treated as the
fourth dimension -- because they then possess a higher symmetry. The development
of theoretical physics over the past century has essentially been based on the
search for the field equations of the forces of nature.

Riemann's metric tensor in
5 dimensions
as expanded by Kaluza (adding a fifth column and row) so
that the 4-dimensional metric of Einstein could be unified with the electromagnetic
field of Maxwell -- unifying the theory of gravity with that of light.

g11

g12

g13

g14

g15

g21

g22

g23

g24

g25

g31

g32

g33

g34

g35

g41

g42

g43

g44

g45

g51

g52

g53

g54

g55

This approach was then extended by Kaluza, as indicated above, to provide a
basis for unifying Einstein's metric with that of Maxwell. Further expanding
the metric tensor in this way subsequently allowed all known forces (gravity,
electromagnetism, weak and strong nuclear forces, and most fundamental particles)
to be integrated into the unified description. Note that by slicing the metric
tensor into its rectangular components, these are respectively descriptive of
particular forces.

Super Riemann tensor
expanded with
the addition to the fifth dimension of supersymmetry to deal with (some)
fundamental particles (adapted from Kaku, 1994)

Gravity
(Einstein)

Light
(Maxwell)

Weak+Strong
nuclear force
(Yang-Mills)

Quarks-leptons
(Matter)

Light
(Maxwell)

.

Weak+Strong nuclear force
(Yang-Mills)

.

Quarks-leptons
(Matter)

.

The question is whether it is fruitful to consider the magic square disposition
of the 81 insights of the Tao Te Ching (in the accompanying
paper) as in anyway corresponding to such a metric tensor. Each numbered
insight would then hold an aspect of the information which -- with that associated
with other numbers -- would define how much the "communication space"
was bent or curved at that point. Recall that the geometry of such curvature
in space-time had been determined by Riemann and Einstein to be indicative of
the forces operating at that point.

Experiencing the forces of unseen connectivity -- mathematically
described

The focus here is on comprehension by the individual in interacting with the
contextual reality at any one moment. The question is whether there is any mathematically-based
conceptual bridge that would clarify the relationship between "geometry"
and "felt forces" in psychological and communication terms -- rather
than in the material terms that are the focus of the metric tensor above.

One insightful approach is that of Arthur
Young who was inspired by his experience in inventing the Bell helicopter,
because of the need for the operator to control movement in three dimensions.
His theory of process is
a formal analytical model based on number theory, geometry and topology -- which
endeavoured to relate to psychologically-oriented modes of knowledge and insight.
Young used this model to help comprehend and integrate a number of disciplines
and areas of inquiry. His original study Geometry
of Meaning (1976), derived from an ordering of 12 dimensionless physical
constants, offers a useful basis for exploring a diversity of issues relating
to learning/action
cycles, dialogue,
sustainable development
and experience of past-present-future
complexes.

The mathematician who appears to have been most helpful in that respect is
Ron Atkin (1972, 1974, 1976, 1977) -- whose ideas he has articulated more accessibly
(Multidimensional man: Can man live in 3-dimensional space? 1981). Atkin
proposed the use of simplicial complexes to analyze connectivity in social systems,
like cities, committee structures, etc. Since then, Atkin's ideas have been
developed further, resulting in a new combinatorial homotopy theory of simplicial
complexes. In this setting, a graded group is associated to a simplicial complex,
similar to the fundamental group of a topological space. However, the resulting
theory is very different from classical combinatorial homotopy theory. Q-analysis
is a combination of geometric and algebraic tools for studying relationships
and connectivity among entities in a complex system. The research generalizes
the idea of binary relation between two things, which underlies the highly successful
theory of graphs and networks. Hypergraphs provide a first extension, allowing
edges with more than two vertices. The methodology of q-analysis extends this
by considering relational structure and multidimensional connectivity. Atkin
was especially interested in traffic on hypergraphs.

Using q-analysis for organizational analysis, in the Management of Technology
Group of the Simon Fraser University (UK), the focus has been on change decisions
and management, which are often the marking points in the life of manufacturing
organizations where such analysis has been explored as a change management tool
that allows the analysis of the change process. The task involved the analysis
of the relationship between various organizational forms in the studied artefacts
and their respective characteristics in order to unearth the connectivity between
various forms. The result of the analysis was then used to assess the change
from one organizational form to another. Keys to success were: (1) confirmation
of groupings, (2) verification of evolutionary pattern, (3) exploration of the
relationships between organizational forms and characteristics sets [more].
This preoccupation with change processes is of course the core focus of the
"sister" classic to the Tao Te Ching, namely the I Ching
(or Book of Changes).

I know of a single daring attempt (which is far from being completed) to
formulate a rigorous mathematical procedure to compute predisposition. It
was made by the British mathematician Ron Atkin (1972). He developed a concept
of connectivity and applied it to such diverse fields as mathematics, politics,
military strategy, chess, regional issues, family therapy, interaction of
atoms and molecule, etc. (Atkin and Johnson, 1992). In the present context,
the merit of Atkin's work is finding the formal language that adequately describes
his concept. The formal constructs, borrowed from algebraic topology constitute
an important step in the mathematical analysis of the problem, including its
application to chess (Atkin, 1972, 1975).

Jacky Legrand (How
far can q-analysis go into social systems understanding?) provides a
detailed critical review of the applicability of q-analysis. She is concerned
at the degree of "metaphorical discourse heavily flavoured by the methods
of algebraic topology, abstract methodology, practical applications and their
relationships" and the need to "separate the syntactic perspective
from the semantic perspective". Her major conclusion is that "the
gap between metaphorical discussion and woolliness is narrow. The understanding
of some of Atkin's ideas has been too intuitive in the past. However the use
of graphics as a language is a powerful thinking tool and Atkin has delivered
a framework for thought".

Possibility of nesting disparate systemic insight sets

Using, by analogy, the method indicated above of expanding the metric tensor,
it is interesting to reflect on the possibility that the psychological "forces"
in "communication space" of which an individual might be consciously
aware -- or be forced to respond to -- could be represented as nested (and "integrated")
in the following way.

.

12

22

32

42

52

62

72

82

92

102

112

122

.

1

4

9

16

25

36

49

64

81

100

121

144

.

/

(Jung types)

/

(Enneagram)

/

Myers-Briggs

/

.

/

.

/

.

/

I Ching

/

Tao Te Ching

/

/

/

/

/

/

/

/

/

.

.

.

Use of the term "communication space" here is misleading -- especially
when it describes the space which may have the range of psychological dimensions
with which psychologists, philosophers and meditators endeavour to come to terms
-- or the space in which the many aphorisms reflecting the wisdom of a culture
are recognized. In addition the experiential nature of that "space"
changes with the dimensionality accorded to it. It is tempting to see the lower
dimensional portions of the above as descriptive of the more "obvious"
forces experienced in daily life, whereas the higher dimensional portions correspond
to various emergent levels of meta-reflection on that experience -- and possibly
as a result of extensive life experience. Confucius is reported to have declared
that only those above 60 years could hope to understand the I Ching !

The difficulty with such mathematical objects is that, aside from their elegant
symmetry, they appear to lack the kind of differentiated content that would
enhance the richness of their significance from a psychological perspective.
They lack engaging "content" and might well be described as sterile
crossword puzzles -- interesting exercises for the mind that do not touch other
psychological dimensions. The content of the 81 insights of the Tao Te Ching
are in stark contrast to this. Each insight has a quality that may be variously
interpreted as description, injunction, orientation, inspiration, etc.

It is perhaps here that the notion of aesthetic associations -- as embodied
in and triggered by music, song, poetry and drama -- might be seen as represented
in some way by the "magic" mathematical properties of such ordering.
This is perhaps most obvious in the mathematics of the particular harmonies
of plucked strings (echoes of Pythagoras!). Are particular insights linked in
a privileged associative manner that is in some way consonant with such properties?
In this sense could the "magic" of a poem be described in terms of
the "magic" properties of some suitable mathematical object -- remembering
that the Tao Te Ching is also considered to be a poem? Is q-analysis
of value in exploring the degrees of association in terms of connectivity? Michio
Kaku (p. 130) makes the point that: " In some sense, the equations of physics
are like the poems of nature. They are short and are organized according to
some principle, and the most beautiful of them convey the hidden symmetries
of nature".

More intriguing is the possibility that the perception of the poem as "magical"
may be dependent on the dimensionality of the "communication space"
within which it is experienced -- such that the "magic" properties
emerge. There is also the question of the ordering of the insights. What are
the mathematical properties of the object used to map the 81 insights? Does
identifying richer "magical" mathematical mappings ensure a more "magical"
aesthetic appreciation of the poem? How do the various mathematical properties
(perfect, semi-perfect, etc) affect the aesthetic properties?

What is the psychological implication of using a magic hypercube instead of
a magic square to order and interrelate the 81 insights of the Tao Te Ching?

The important point to be made here is, as in mathematics, that "aesthetic"
may have powerful additional implications. For mathematics the stress is on
the theoretical implications of the beauties of symmetry as an indicator of
"correctness" and "power of explanation". In the psychological
case it is associated with integrative, memorable insights -- binding together
patterns of insights in resonant forms that provide the foundation for the emergence
and comprehension of higher forms of order, and the potential for identification
with it in some enhancing manner. "Aesthetic" configures insights
in such a way as to sustain the emergence of a higher, subtler form of psychological
identity.

Helpful research in this direction is that of George Gadanidis and Cornelia
Hoogland (Mathematics
as story, 2002) who contend that mathematics is an aesthetic and a storied
experience. They explore the interplay between what is an 'aesthetic mathematics
experience' and a 'good mathematics story' using a mathematical applet, namely
a Colour Calculator -- that resembles the magic squares discussed in
the accompanying paper.

Navigating the psychological forces of "communication
space"

The chess board is often used as a metaphorical descriptor of the metric tensor
and of magic squares. In playing chess or go, there is a real sense of
"forces" of different quality and potential "in play" across
the board -- as in any drama. In response to the play of forces in a complex
social world, some are recognized as "skilled operators". Curiously
some of the interactions are recognized as "doing a number" -- presumably
deriving from numbered tactics in team sports. Thierry Gaudin (Les katas institutionnels.
Transnational Associations, 30, 1977, 3,pp. 77-79) identified 21 tactical
moves (katas) open to institutions.

The term "field" has not only been applied to describing physical
nature, but also to understanding of psychological and social natures. Gestalt
psychologists (Köhler, 1942) have conceptualized a psychic energy field under
stress and with forces tending toward sensory and cognitive unity and balanced
simplicity. Influenced by Gestalt psychology, Kurt Lewin (1951) in his psychological
field theory thought of psychic energy localized in systems of tension and forces.
Needs generate the field within which our potential activities and goals become
manifest. Following Lewin, Edward Tolman (1951) considered sensory and cognitive
psychological elements as affected by need-push forces activated in an energy
system. Sociologists also have used field in this meaning. B. F. Brown (1936),
a student of Lewin's, considered social behavior a result of individual needs
localized in energy systems of tension and forces [more].

Studies of the origins of chess-type games in different cultures, including
9x9 variants, emphasize the ways in which they reflect the psycho-social forces
in play (see Pavle Bidev. Chess:
a mathematical model of the cosmos, 1979; Ricardo Calvo. Ancient
Gnosis and Chess Evolution, 1999 and Continued
Extracts on Gnostic Elements in Chess, 1999). Ricardo Calvo concludes
that "The movements of the pieces are based in mathematical considerations
that are older than the game of chess itself" [more].
Of special intedrest are the cross-cultural anthropological studies of the phylogentic
relations between some 40 chess-type games, their connection with divination
and the development of the magnetic compass, and the relationship between the
current 8x8 variant and the 9x9 variants -- such as the extant Xiangqi
(Chinese) and Shogi (Japanese)
(see Gerhard Josten. Chess:
a living fossil, 2001). It has been argued -- notably in the light of
the jumping Rook, together with the movements of all pieces of Chaturanga
as seen in the numerical arrangement of a magic square of 8x8 (the so-called
Safadi Board)
-- that the chess movements were historically deduced from a "genetic code"
of arithmetical operations.

Knight's move: In chess, this is especially interesting given the potential
significance of the moves of the knight -- as a "noble" rather than
as a "commoner". The strangeness of the knight's move (a keima
in the Japanese game of go), and its numerical symbolism, has traditionally
been the focus of hypotheses connecting the origins and structure of chess with
secret magical and religious rituals of ancient India.

In their study of its significance, James E. Loder and W. Jim Neidhardt (The
Knight's Move: the relational logic of the spirit in theology and science,
1992) focus on the expression of complementary thinking that facilitates positive
interaction between science and Christian theology. A reviewer, Richard H Bube
(The "Strange
Loop" of Complementarity) notes: "The symbol of 'the Knight's move'
refers to the unique move of the chess piece that is the only one not moving
in a straight line, as an indicator of a leap of insight or a leap of faith.
The book also draws heavily on the symbolism of the Moebius strip, the two-dimensional
'strange loop' twisted in the middle, which has a two-dimensional surface that
can be totally traversed with continuous motion along the strip".

The problem of the knight's tour [more]
on traditional 64-board in chess was solved by Euler in 1759. Knight's tour
and knight's path are special cases of Hamiltonian cycles and Hamiltonian paths
in graph theory [In August 2003 it was announced
that one of the classical unsolved problems of mathematics, concerning the existence
of a path that could be traversed by a knight on an empty numbered 8 x 8 chessboard,
had been proven to be without solution].

The move of the knight is used as a metaphor for the unexpected, and illogical,
connections between ideas -- invisible to the "commoner". Sidney Cohen
described LSD perception as a kind of knight's-move thinking which leaps over
logical premises and formal syllogisms. "Knight's
move thinking" is even considered a pathological condition of thought
disorder denoting a lack of connection between ideas -- an illogicality of the
loosening of associations (found in schizophrenia but to be contrasted with
the flight of ideas which characterizes hypomania). Strategically it is appreciated
as an out-flanking maneuver. [The knight is part of the emblem for the US
Psyops as a traditional symbol of special operations signifying the ability
to influence all types of warfare.]

The knight's move can be used to illustrate how "innovation" can
emerge from a point W (below). Whereas the "logical", "linear"
moves from W are along any of the grey pathways (whether horizontally, vertically,
or diagonally), the knight can move outside this logical framework, first to
X, then to Y. In a sense the originality or novelty associated with X is "birthed"
by the vertical and diagonal pathways from W. What is "birthed" is
in a sense hidden from the linear outlook along the grey pathways from the W
perspective.

Y''

Y'

X'

X''

Z''

Z'

W

The mathematical concern with the knight's tour might perhaps be usefully explored
in relation to the cultural Grand Tour considered appropriate to the education
of nobility of the 18th century -- through which they learned about the politics,
culture, and art of neighboring lands. Psychologically it was an exploration
of different realms of "communication space" -- distinct from those
accessible through the logical framework of the point of departure.

Other moves: One of the merits of board games, such as chess and go,
is their capacity to give people a sense of the psychological significance of
other moves. One of the merits of certain Eastern martial arts, such as aikido,
is to extend this to more complex dimensions of communication space. As Clifford
Pickover (The Zen of Magic Squares, Circles, and Stars, 2002) has noted,
since the dawn of civilization humans have invoked such magical patterns to
ward off evil and bring good fortune -- yet who would have guessed that in the
twenty-first century, mathematicians would be studying magic squares so immense,
and in so many dimensions, that the objects defy ordinary human contemplation
and visualization?

String theory and modular functions

The scope of the unification in physics achieved by the above generalization
of field theory proved inadequate (given the exclusion of certain fundamental
particles) to the challenge of a complete "theory of everything" --
integrating space-time and matter. String theory, with its unusual geometry
-- strings vibrating self-consistently in 10 or 26 dimensions -- proved to be
the missing link. According to Michio Kaku (co-founder of string theory): "The
distinguishing feature of a string is that it is one of the most compact ways
of storing vast amounts of data in a way in which information can be replicated"
(p. 156). And: "The symmetries of the subatomic realm are but the remnants
of the symmetry of higher-dimensional space" (p. 159). As remnants they
emerge from the curling up of that space -- as with such visible symmetries
as rainbows and crystals.

A major concern for physicists is why string theory is defined self-consistently
in only 10 or 26 dimensions. The explanation is associated with the modular
functions identified by Srinivasa
Ramanujan (1887-1920) and named after him. As Michio Kaku explains:

When the Ramanujan function is generalized, the number 24 is replaced by
the number 8. Thus the critical number for the superstring is 8+2 or 10 [adding
two dimensions for the case of relativistic theory]. This is the origin of
the tenth dimension. The string vibrates in ten dimensions because it requires
these generalized Ramanujan functions in order to remains self-consistent.
In other words, physicists have not the slightest understanding of why
ten and 26 dimensions are singled out as the dimension of the string.
It's as though there is some kind of deep numerology being manifested in these
functions that no one understands. It is precisely these magic numbers appearing
in the elliptic modular function that determines the dimension of space-time
to be ten. (p. 173, italics in original).

Despite this, physicists remain mystified as to why such magic numbers emerge
so definitively. The 10-dimensional theory of hyperspace remains untestable
and Michio Kaku (p. 179) asks the question: "Is beauty, by itself, a physical
principal that can be substituted for the lack of experimental verification?"

But, relevant to the possible higher integration within the Tao Te Ching,
Michio Kaku (p. 172) acknowledges that as mysterious as are the modular functions
was the self-taught Ramanujan: "...the strangest man in all of mathematics,
probably in the entire history of science. He has been compared to a bursting
supernova, illuminating the darkest, most profound corners of mathematics...".
The incredible theorems in number theory that he exuded -- "half a dozen
new ones, almost every day" -- have aroused wonder at the unconventionality
of his thinking processes. He has been described as intuition incarrnate (Robert
Kanigel, 1991). He is estimated to have produced between three and four thousand
theorems -- as many as two-thirds being new to mathematics [more].

For Jonathan Borwein (Ramanujan and Pi. Scientific American,
February 1988, p. 112): "He seems to have functioned in a way unlike anybody
else we know of. He had such a feel for things that they just flowed out of
his brain. Perhaps he didn't see them in any way that's translatable."
For Ramanujan, they emerged from his "dreams" inspired by the Hindu
goddess Namakkal.

It is curious that, of the mathematicians acknowledged to be the greatest of
all time (Archimedes, Euler, Gauss, Jacobi, Newton and Ramanujan), both Ramanujan
and Newton were inspired in ways which are considered so irrational that they
are an embarrassment to their professional peers. How did Ramanujan's "dreams"
work -- given that he believed them to be the inspiration of a goddess? Why
was the, spiritually inspired, Newton's work on alchemy -- that he believed
to be fundamental to his understanding -- considered irrelevant to his mathematics?
Are such eccentricities to be equated with substance abuse as incidental to
mathematics -- or do they have a role in "Knight's move thinking"?
How does intuition work?

Cosmology -- Big Bang to Big Crunch

As explained by Michio Kaku, introducing the higher dimensions of hyperspace
may also be essential for prying loose the secrets of creation. For:

According to hyperspace theory, before the Big Bang, our cosmos was actually
a perfect ten-dimensional universe, a world where interdimensional travel
was possible. However, this ten-dimensional world was unstable, and eventually
"cracked" in two, creating two separate universes: a four- and a
six-dimensional universe. The universe in which we live was born in that cosmic
cataclysm. Our four-dimensional universe expanded explosively, while our twin
six-dimensional universe contracted violently, until it shrank to almost infinitesimal
size...The energy that drives the observed expansion of the universe is then
found in the collapse of ten-dimensional space and time. (Hyperspace,
1994, p. 27)

The current expansion of the four-dimensional universe is eventually expected
to go into reverse -- leading to the Big Crunch. Physicist Gerald Feinberg speculated
that the one way for intelligent life to avoid this final calamity would be
through mastering the secrets of higher-dimensional space -- so that in the
final moments of collapse, intelligent life forms may be able to tunnel into
high-dimensional space or an alternative universe. The language recalls that
of those focused on the rapture of the "end times" Biblical scenario.

From a psychological perspective this concept might be interpreted as an effort
to project as far as possible from the present -- into the most inaccessibly
distant past -- a "golden era" of integration. And as an effort to
project into the inaccessibly distant future -- the possibility of re-integration.
This may be consistent with the continuing depersonalized globalization of the
world of material value according to a constrained logic -- as matched by the
continuing collapse of individual spiritual life, forced to "curl up"
into insignificance. It is perhaps no wonder that the importance of drugs and
substance abuse is increasing explosively to offer individuals access to "knight's
move thinking" (see above) with its more creative freedom of association.

But from a "psycho-spiritual" perspective, it is also interesting
to speculate on the possibility that the "communication space" experienced
by an individual is subject to an analogous explosive expansion at birth --
and to violent collapse at death. Or, even more intriguing, that such an analogous
explosive expansion takes place in any significant moment of creativity in the
life of an individual -- to be lost (or quashed) with any subsequent reversion
to banality or loss of focus (or meditative concentration). This might accord
with some existential and meditative experiences which -- as with many high-energy
physical experiments -- would be difficult to demonstrate or replicate.

Higher dimensionality as the prime characteristic of human
consciousness?

Physicists make much of how inaccessibly small (10-33cm, the Planck
length) is the curled up fifth-dimensional space from which humanity is divorced
-- and of how much energy would be required to demonstrate its existence. There
is an ultimate irony to the possibility that it is precisely this infinitely
small high-dimensional space that is in some way the locus of what scientists
have been unable to locate inside the body -- namely whatever constitutes "life"
or, in its higher dimensions, "soul" or "spirit". Humans
may have a much more intimate relation to such higher dimensions -- in fact
this intimacy may be precisely what characterizes consciousness and the sense
of selfhood.

It is curious that mathematicians do not wonder at their capacity to wonder
at their understanding of such high dimensionality (Kaku, p. 214). Is it not
possible that such understanding derives from a degree of correspondence (or
resonance) between the dimensionality of the comprehending mindset and that
of what it frames as comprehensible?

Is it possible that humans are essentially six-dimensional entities functioning
within a four-dimensional space -- or trapped by a mistaken self-image as four-dimensional
entities? (see Metaphoric
entrapment in time, 2000). If light is indeed to be understood by physicists
as a vibration in the fifth dimension (a warping of the geometry of higher-dimensional
space), what of the sense that someone is "brilliant" (as many physicists
are perceived to be, especially Ramanujan) or "enlightened" (as some
gurus are perceived to be) or "charismatic" (as with some media personalities)
-- others are perceived as "dead", or without a "spark of life".
What is the dimensionality of the space within which "good vibes"
are detected -- or the "gravitas"
so important to political success? Such points would correspond to the argument
of physicist Peter Freund that the reason the forces of nature appear so fragmented
in three-dimensional laboratories is that, like a miserably caged cheetah unable
to run, their true home is in higher-dimensional space-time (Kaku, p. 12).

However the concern is to be framed, there remains the possibility that exploration
of higher dimensionality offers a means of "escaping" from the catastrophe
towards which many perceive life on the planet to be heading in the not too
distant future. The question is how to resolve dangerous incompatibilities of
perspective driving this tendency by shifting to a perspective of higher dimensionality
-- so as to integrate "differences" that manifest so disastrously
at lower dimensionality. Not only may higher dimensionality be the ultimate
source of unity in the universe (Kaku, p. 15), it may also offer the potential
for unity in a globalized psycho-social system subject to self-destructive fragmentation.

Resonant pattern of associations

As implied above, it would seem that the central psychological dimension of
this argument can be developed through the notion of "associations",
semantic or otherwise, whether seen in terms of the poetic beauty of their symmetry
or through the connectivity that they mark. The emphasis to date appears to
have been on simple, or first order, semantic associations which are the immediate
priority in information retrieval in "second
generation knowledge management" and the envisaged "semantic
web" [more
| more
| more
| more].
Higher order associations, as now explored by "latent
semantic analysis" and "high dimensional conceptual space",
are another matter [more
| more].

In English poetry, the single most compelling discriminator of that genre
-- that which defines a poem as a poem -- has traditionally been its meter.
Meter defines the length of the line, and thus the distinctive look of a poem
on the page, and it sets, for the hearer of a poem, the telling regularity
of a rhythm. Whether this rhythm also carries the burden of some of a poem's
meaning or whether it is used only for a conventional aesthetic effect that
invites the reader to take pleasure in its regularity or variations, meter
is one of the central attributes of the genre of poetry.

Magic numbers, or the symmetry of magic squares, point to degrees of organization
of tantalizing significance. They may mark mnemonic highways -- like those described
in the methods of calculating prodigies for whom numbers "call to one another"
in unusual ways. They may be understood as the fine structure of what Gregory
Bateseon (Mind and Nature: a necessary unity, 1979) famously described
as "the pattern that connects". The question of this paper is whether
there is a higher order "pattern that connects" (possibly an "overtone")
interweaving the 81 insights of the Tao Te Ching.

Static vs Dynamic: Associations may easily be understood in static terms
only -- as networks of various kinds (kinship, information exchange, etc). But
they may also be understood more fruitfully as dynamic relationships -- as "resonant"
-- suggesting analysis in terms of vibrations, as in string theory. Again, if
light can be understood as a vibration in a higher-dimensional space, then perhaps
this is also the case with aesthetic and mytho-poetic associations. And, like
beauty, perhaps humour is a vibration characteristic of certain symmetries in
semantic hyperspace -- evoking laughter as a vibration at another level. This
would be consistent with some Zen and Taoist "crazy wisdom" perspectives.
It also points to the ambiguity of "a-musing" -- as an inhibitor of
a muse or as a characteristic of its activity.

Sense of identity: It might be asked to what extent a sense of personal
(or collective) identity can be understood as a pattern of resonant associations.
This possibility is indicated, for example, by the manner in which composers
are recognized through their music (through their subtleties of "style")
-- even in the case of unknown compositions. Would many musicians not prefer
to be identified and remembered through the harmonies of their melodies and
compositions -- a musical identity or creative style -- rather than through
a name (see Raymond A. R. MacDonald, et al (Eds). Musical
Identities, 2002 [extract])?This musical metaphor can be extended to the "comoposition" of
a person's relationship with the world in daily life (see also Mary Catherine
Bateson. Composing a Life. 1989) although the directive dimension of
composition (by others in the future) needs to be complemented by "interpretation",
"improvisation" and "performance" (play in the moment).
Or, using a poetic metaphor, as indicated by Wallace Stevens:

"The subject matter of poetry is not that 'collection of solid, static objects
extended in space' but the life that is lived in the scene that it composes;
and so reality is not that external scene but the life that is lived in it."
(The Necessary Angel, 1951)

Identity could indeed be understood in terms of self-consistency and degree
of connectivity. What is lost when a person is described as "losing it"?
How is the classical philosophical and spiritual question -- "who am I"
-- to be understood in relation to identity established in terms of higher-dimensional
connectivity?

Higher dimensional identity is not something reserved for mathematical or creative
geniuses -- or for the spiritually enlightened. It manifests variously in maturity,
gravitas, humour, charm, etc that are beyond the conventional socio-economic
and psychological definitions of humans. It is only "distant" or "insignificant"
to the extent that the focus is on the four-dimensional material reality of
space-time. Why are some songs widely popular as carriers of the Zeitgeist
of a period? Do they in some way sustain a pattern of identity? Can the personality
type frameworks of Jung, through Myers-Briggs for example, be usefully reframed
in terms of patterns of connectivity across a magic square as suggested by John
C. Gonsowski (Personality,
Physics and Spirituality: a common geometry, 2001)?

Creativity and originality: muses and rasas

The 9-fold organization of the Tao Te Ching was explored in terms of
magic squares (in the accompanying
commentary). It might be asked whether there is any possibility that it
may also have a 9-fold organization, from an aesthetic perspective, that might
in some way be consonant with the mathematical ordering. For a creative person,
especially in the arts, the psychological significance of an inner "muse"
may well be perceived as essential. A muse might be understood as a catalyst
for the connectivity of the "pattern that connects". There is a case
therefore for looking at the variety of muses -- presumably eliciting different
kinds or qualities of connectivity. In classical Greece and Rome, 9 such goddesses
were identified as sources of inspiration in the arts and sciences (see Angeles
Arrien. The Nine Muses: a mythological path to creativity. 2000).

Muse name

Muse domain

Calliope

Epic Poetry

Clio

History

Erato

Love Poetry

Euterpe

Music

Melpomene

Tragedy

Polyhymnia

Sacred Poetry

Terpsichore

Dancing

Thalia

Comedy

Urania

Astronomy

Whilst the nine muses are identified here in terms of aesthetic form,
a classical Indian analogue, the nava rasa (or nine sentiments), emphasizes
the aesthetic quality in the performing arts (music, dance, drama or
poetry) that colour the mind with a particular feeling, sentiment, passion or
emotion.

Rasa name

Rasa domain

shringara

love, sensual, romantic, erotic

hasya

humour, happiness

karuna

sympathy, compassion, sadness

raudra / krodha

anger, fury

veera / viraam

heroism, courage, majesty

bhayanaka / bhaya

fear, terror

vibhatsaya / bhibasta

disgust

adabhuta

wonder, amazement, surprise

shanta

peace, serenity, tranquility

Presumably, in some way, such qualities are essential to the experience of
the field of consciousness for which the Tao Te Ching supplies marker
insights. In music, for example, the structure of each raga -- the main
form of Indian classical music -- and the melodic movement within its framework,
are governed by definite and extensive rules. The technique of a raga
consists in the use of certain fixed notes and microtones to the deliberate
exclusion of others. Within this fixed framework, however, there is unlimited
scope for improvisation. Rasa literally means juice (recalling the concern
with the flow of "creative juices" along certain pathways), but in
a musical context, it refers to the mood or sentiment created by a raga.
In theory, if every permissible permutation and combination of notes was exploited,
it would yield 38,000 ragas -- as it is only about 200 are common [more].

Ramanujan articulated the influences on his thinking explicitly in terms of
a goddess, Namakkal, that was the source of his "dreams". Ramanujan's
sense of self-identity cannot be usefully said to be centered in four-dimensional
space, when his life's preoccupations were of higher dimensionality. In his
case, the muse is the catalyst for access to the "music of the spheres".
It is the artist's muse that sustains the pattern of connectivity -- the "semantic
music" -- essential to creativity. In its absence, the artist is left bereft.
Connectivity may in this sense be understood as coming and going -- rising and
falling like a tide -- as access to higher dimensionality is gained or lost.
One of the charms of learning is acquiring access to such connectivity -- enculturation
-- later to be followed by the tragedy of its dissolution ("losing it")
with loss of memory and senility.

What indeed is creativity in this context? How might it be defined in terms
of mathematical discovery of new patterns of associations -- engendering "semantic
flowers" as attractors? From where do they emerge? The term "originality"
can also usefully imply a return to an "origin" -- suggesting a sense
in which this might, in some measure, be a return to the perspective of 10-dimensional
integrity from which perceptible patterns of symmetry emerged.

An aesthetic perspective opens the possibility of the different rasas (or muses)
being evoked in various combinations. In the case of the Indian tradition --
through the legend of the Vastu
Purusha Mandala -- the 9-fold organization of aesthetic quality is fundamental
to marking out a space of 9x9 squares that is the basis of Vastu Vedic architecture
and design, notably in defining a courtyard. This might be understood as the
projection of a higher dimensional order onto a two-dimensional pattern open
to experience. Other square patterns are also used [more]
to optimize "energy" distribution according to what is effectively
the Vedic equivalent of feng shui (see also B B Puri. Vastu Science
for 21 Century, 2003). Such spaces may be walked and experienced in ways
that recall early western use of mnemonic architecture (see Frances Yates. The
Art of Memory, 1966).

Aesthetically, an individual ("X" in the table below) might then
be understood as subject to a confluence (or configuration) of influences in
exploring the squares of the space (as indicated, at one moment, by an assumed
combination of rasa "mind colourings" in the table below).
A given influence might be reinforced if it was conveyed directly through contiguous
squares (whether vertically, horizontally, or along a diagonal) and inhibited
in the absence of any such direct connectivity -- as illustrated by the coloured
squares below. Any "magic square" properties to the confluence would
then enhance such influences -- through bringing to bear secondary aesthetic
associations (not shown). The space may of course be 2-dimensional in a physical
walk, or multi-dimensional in the case of a mind-walk.

...the vastu-purusha mandala is an image of the laws governing the cosmos,
to which men are just as subject as is the earth on which they build. In their
activities as builders men order their environment in the same way as once
in the past Brahama forced the undefined purusha into a geometric form....
building is an act of bringing disordered existence into conformity with the
basic laws that govern it. This can only be achieved by making each monument,
from the hermit's retreat to the layout of a city, follow exactly the magic
diagram of the vastu-purusha mandala.

A pattern of associations, whether in a conceptual scheme alone or embodied
in architecture, might usefully be understood as a receiver of energy -- functioning
like an aerial array in response to vibrations of higher dimensionality -- whether
spiritual inspiration or the insights received by Ramanujan. The vibrations
of music and song may in this way act as a source of invigorating energy, nourishment
and coherence.

Michio Kaku (p. 11) argues that the incapacity of humans to visualize four
dimensions is due to lack of selection processes to develop such skills -- in
the absence of people needing to respond to "lions and tigers" leaping
at them out of the fourth dimension. It could however be argued that other responses
have been developed to higher dimensional "wildlife" -- but these
are not associated with the sense of sight. He notes (p. 12) that the laws of
nature become simpler and more elegant when expressed in higher dimensions,
but it may also be the case that the appreciation of the wilder harmonies of
nature -- by poets and deep ecologists, for example -- derive from senses unrelated
to the three-dimensional world.

Interdimensional travel: Higher dimensionality may be evident in the
resonant associations characteristic of a pattern of sustained and skillful
repartee within a group -- beyond the formal, rule-bound constraints of verbal
tennis. The thematic interplay on multiple semantic keyboards suggests a facility
of transition -- through semantic wormholes -- beyond disparate semantic universes.

Michio Kaku explores the future possibility of inter-dimensional travel --
wormholes between parallel universes -- once facility with the six curled up
dimensions is developed. How are the skills of transdisciplinarity -- and of
a polymath -- to be compared with this? Might the challenge of interdisciplinary
and intersectoral thinking be clarified if framed in terms of opening semantic
wormholes? (see also Transdisciplinarity-3
as the emergence of patterned experience1994). Is it possible that,
in contrast to conventional text, knowledge will in future be organized and
hyperlinked on the basis of some kind of higher dimensional "magic square"
pattern enhanced mnemonically by aesthetic associations (see, for example, Structuring
Mnemonic Encoding of Development Plans and Ethical Charters using Musical Leitmotivs,
2001)?

For those prepared to set aside physical attributes, social status and pheromones,
how then to understand the nature of mutual attraction and love (or antipathy)?
To what extent is it a form of higher dimensional pattern matching and "goodness
of fit"? To what extent might people be understood to function as "stargates"
-- pulling each other across semantic hyperspace (see People
as Stargates: an alternative perspective on human relationships in space-time,
1997)?

Time travel: The possibility of time travel, in the light of hyperspace
theory, is also explored by Michio Kaku. The nature of time travel from a psychological
perspective is perhaps usefully indicated by the classic verse of the poet T
S Eliot: "We shall not cease from exploration / And the end of all our
exploring / Will be to arrive where we started / And know the place for the
first time" (Little
Gidding, 1943). A cyclic return to the point of origin.

As Kaku (p. 95) notes with reference to the experience of living in hyperspace:
"The counterintuitive stunts that are possible in a hypersphere are physically
interesting because many cosmologists believe that our universe is actually
a large hypersphere." But Eliot's poetic account may apply even more appropriately
to the six-dimensional "curled up" framework which may be the nature
of the individual cognitive environment of humans.

Antonio de Nicolas offers a powerful insight into such time travel in his poetic
exploration of Remembering the God to Come (2000) -- whose title emphasizes
the cognitive role in (re)assembling a degree of integration in the light of
implicit knowledge of its nature. Curiously, it is in an alternative community,
Damanhur, that the most active experiments
in time travel are undertaken -- in part dependent on appropriate psychological
training (see Timeship:
Conception, Technology, Design, Embodiment and Operation, 2003)

Cultivating the moment

As a philosopher, Antonio de Nicolas (1978) points to the role of music or
song, embodied in the moment, to engender coherence in the present. He uses
the non-Boolean logic of quantum mechanics of Patrick Heelan (The logic of
changing classificatory frameworks, 1974) to explore the epistemological
significance of cognitive experience grounded in tone and the shifting relationships
between tone in the Rg Veda. For him, it is through the pattern of musical
tones that the significance of the Rg Veda is to be found:

"Therefore, from a linguistic and cultural perspective, we have to be aware
that we are dealing with a language where tonal and arithmetical relations
establish the epistemological invariances... Language grounded in music is
grounded thereby on context dependency; any tone can have any possible relation
to other tones, and the shift from one tone to another, which alone makes
melody possible, is a shift in perspective which the singer himself embodies.
Any perspective (tone) must be "sacrificed" for a new one to come into being;
the song is a radical activity which requires innovation while maintaining
continuity, and the "world" is the creation of the singer, who shares its
dimensions with the song." (1978, p. 57)

This offers new dimensions to sacrifice that contrast with those required by
contemporary economics.

The physical effects of resonance from sound and vibration are well known (for
example, Chladni patterns in two
and three
dimensions [more]).
Can psychological analogues be set up to engender the future and exert a time-binding
force? Meditation on yantras and mandalas would seem to have a related function
-- traditionally linked to the magic squares discussed in the accompanying
paper. Within such a context, can analogues to overtones function as vehicles
for particular forms of understanding? As indicated there, interesting patterns
can be generated from magic squares when the numbers of the squares are replaced
by symmetric symbols. These resemble Chladni patterns. Whether the magic square
(or higher dimensional) patterns can be more readily comprehended through use
of auditory display techniques (see NSF The
Sonification Report), as seems highly probable, remains to be discovered.
This could be a valuable way to explore and navigate comprehension of the relationships
between the 81 insights of the Tao Te Ching -- especially in the light
of any insights concerning Indian rasas (see above). Lars Kindermann's
downloadable MusiNum: The Music in the
Numbers software -- and its experimental use in the Elenyscope
-- is an interesting step. [See very preliminary experiments in separate paper
Musical Articulation
of Pattern of Tao Te Ching Insights: Experimental sonification based on magic
square organization, 2003]

There is a poetic irony to the manner in which this argument suggests that
the appropriate relationship to experience in the moment is effectively through
its "cultivation" as a field of consciousness -- given Faraday's original
use of the agricultural "field" metaphor as a basis for the field
theory that has been so fundamental to hyperspace theory. Other valuable insights
may be associated with such metaphoric correspondence in exploring the psychological
dimensions of the global concern with sustainability on a planet facing catastrophe
(see Psychology
of Sustainability: Embodying cyclic environmental processes, 2002).

En-minding the extended body?

From a cognitive perspective, the dynamic between the four-dimensional world
of space-time and the six-dimensional world, within a ten-dimensional framework,
can be usefully explored in the light of enactivism,
as developed by Francisco Varela and colleagues (Natalie Depraz et al. On
Becoming Aware: a pragmatics of experiencing, 2003), notably (Laying
Down a Path in Walking, 1987) and with others (The Embodied Mind: cognitive
science and human expression, 1991). In this light, the question of how
an individual might sustain a resonant pattern of associations in relation to
a natural environment -- in whose (re)definition he or she is continually engaged
-- is explored separately (En-minding
the Extended Body: Enactive engagement in conceptual shapeshifting and deep
ecology, 2003)

In the light of the above, Feinberg's speculation regarding the possibility
of intelligent life surviving by tunnelling through to the six-dimensional context
in the final phases of the Big Crunch (when such access becomes physically more
feasible), can usefully be reframed as a far-from-distant possibility. Aside
from the possibility of a purely religious framing (conversion, rapture, "end-times"
scenarios, and the like), the analysis has the quality of a somewhat artificial
polarization, or contradiction, recalling the centuries of "body vs mind"
debate. It ignores other, and more intimate, ways of exploring the nature of
death and the framework within which it is understood to take place.

The Romantic poet-philosopher Novalis
(1772-1801) is merely one of many to argue that: "To annihilate the principle
of contradiction is perhaps the supreme task of higher logic". But for
Hegel: "...the life of the mind is not one that shuns death, and keeps
clear of destruction; it endures its death and in death maintains its being.
It only wins to its truth when it finds itself in utter contradiction".

Such perspectives could be considered natural within the field of insights
of the Tao Te Ching, which -- at one level -- encourages an unseemly
search by some Taoists for forms of immortality inconsistent with its own highest
insights (as with the preoccupation of other religions with a "place in
Heaven"). This ignores higher orderings of those insights that may legitimate
the kinds of "crazy wisdom" perspective of Taoists and Zen (see, for
example, Perle Besserman, Manfred Steger. Crazy Clouds: Zen Radicals, Rebels
and Reformers, 1991). Physicists have their own take on the need for such
"craziness", as illustrated by the much-quoted statement by Niels
Bohr in response to Wolfgang Pauli: "We are all agreed that your theory is crazy.
The question which divides us is whether it is crazy enough to have a chance
of being correct. My own feeling is that is not crazy enough." To that Freeman
Dyson added:

"When a great innovation appears, it will almost certainly be in a muddled,
incomplete and confusing form. To the discoverer, himself, it will be only
half understood; to everyone else, it will be a mystery. For any speculation
which does not at first glance look crazy, there is no hope!" (Innovation
in Physics, Scientific American, 199, No. 3, September 1958)

Again there may be a poetic irony to the secret of immortality buried (à
la Umberto Eco) in the word itself -- whether "I'm-mortality",
or the "I-mortality" for which many mystics quest. Within any more
highly ordered array of insights marked out by the Tao Te Ching, for
example, the key to such "I-mortality" is well put by Reshad Feild
in Sufi terms as the cognitive challenge of "removing the point" from
which one views. Higher mathematical orderings of the 81 insights of the Tao
Te Ching are a powerful guide to doing so.

Summary

The argument in this paper may be understood as a succession of steps:

Complete set of insights: The 81 insights of the Tao Te Ching
are considered as a complete and comprehensive set which is a challenge to
comprehension.

Classification: Consideration of the possibility of the classification
of such insights, notably in tabular or matrix form, to obtain some sense
of pattern.

Patterning across categories: The possibility that relationships
between insights may constitute a pattern of associations of higher order
than the simpler implications of any matrix. Insights from the mathematics
of magic squares can be used as one tool for suggesting a possible basis for
such a pattern.

Aesthetic image: There is a real challenge to deriving integrative
meaning from a pattern of association between seemingly disparate insights,
especially if mnemonic qualities are required. The power of appropriate aesthetics
may be used to enhance comprehension of complex patterns. Distinct semantic
"threads" may then be fruitfully woven together -- perhaps a "magic
carpet".

System: The operational significance of a pattern of associations
also needs to be clarified as a system of interacting forces to which a person
is subject -- or within which a person can act. This dynamic, interactive
perspective can notably be explored through the sense of possible moves in
games (as in chess).

Resonance: The systemic forces can be understood -- and even experienced
-- as vibrations within some form of field which may have qualitiative attributes.
Such vibrations may interact to provide unexpected resonance effects that
can also be appreciated aesthetically.

Emergent order: Resonance effects between insights can be understood
as resulting in emergent forms of higher order ("overtones") --
namely semantically subtler forms of meaning.

Enactivism: The apparently external reality of any pattern of resonance
amongst insights may be understood as engendered by the perceiver ("Laying
down the path through walking"). A diversity of alternative patterns
may be generated in this way.

Identity: Engagement with engendered patterns of order enables a
new sense of identity -- subtler to the extent that the pattern is of higher
order.

8: Enactivism

1: Insights

6: Resonance

3: Patterning

5: System

7: Emergence

4: Aesthetics

9: Identity

2: Classification

It may that the distribution of the above steps within the framework of the
Lo Shu magic square suggests clusters of meaning -- by row, by column or by
diagonal.

Mondo Secter. The 'Zhou Yi" Hexagram Sequence: an authentic, intended binary
system discloses a rational, bilateral mathematical symmetry (Paper for the
8th International Conference on the History of Science in China) [abstract]

Mondo Secter. Locating the Self in the No-Self: The I-Ching as Metaphor for
Six-Dimensional Reality. (Paper for the Society for the Conference on the Anthropology
of Consciousness, 1997 Berkeley, California)

Eugene Stivers and Susan A. Whellan (Eds.). The Lewin Legacy: Field Theory
in Current Practice. Springer-Verlag, 1986 (Recent Research in Psychology)

Gabriel Stolzenberg. Can an inquiry into the foundation of mathematics tell
us anything interesting about mind? In George Miller and E. Lenneberg (Eds).
Psychology and Biology of Language and Thought. New York, Academic Press, 1978